Cumulative Exposure From Multiple Markers

Why Cumulative Exposure Can Matter

A marker measured near an event time may not summarize the history most relevant to risk. For example, a hazard may relate to the amount of a biomarker accumulated over several months, rather than only its value today. With multiple longitudinal markers, this idea can be expressed by constructing a separate cumulative exposure for each marker and including those exposures together in the survival hazard.

The exposure terms are computed from the model-implied latent marker trajectories, not by simply adding observed measurements. That distinction matters: observations are noisy and may be unequally spaced, whereas the latent trajectory represents the marker level over the full time window. The earlier material on the multivariate linear mixed model supplies these trajectories and their subject-specific random effects.

Definition: Let $m_{ik}(s)$ be the latent trajectory for subject $i$ on marker $k$, where $k=1,\ldots,K$. For a prespecified window length $W>0$, define the rolling cumulative exposure at time $t$ by $$ A_{ik}(t;W)=\int_{\max(0,t-W)}^{t}m_{ik}(s)\,ds. $$ When $t\geq W$, this integrates the marker over the full trailing window $[t-W,t]$. When $0\leq t<W$, it integrates from baseline to $t$, so the early-follow-up exposure covers less than $W$ time units.

The window length and its boundary rule are part of the model specification. The definition above uses a truncated window near baseline; it does not assume that a marker was measured or modeled before baseline. A different scientific question might require an exposure accumulated from baseline, or a full window available only after $t\geq W$. Those choices are not interchangeable and should be stated explicitly.

Putting Several Exposures in the Hazard

Let $h_i(t\mid\mathbf b_i)$ denote subject $i$'s hazard at time $t$, conditional on the subject-specific random effects $\mathbf b_i$. A proportional hazards model with $K$ marker-specific cumulative exposures can be written as

h_i(t\mid\mathbf b_i) = h_0(t)\exp\left\{ \boldsymbol{\gamma}^{\mathsf T}\mathbf w_i+ \sum_{k=1}^{K}\alpha_k A_{ik}(t;W) \right\}.

Here $h_0(t)$ is the baseline hazard, $\mathbf w_i$ is a vector of included covariates, and $\boldsymbol{\gamma}$ and $\alpha_k$ are association coefficients. The hazard is conditional on the latent trajectories through $\mathbf b_i$. Holding other terms fixed, increasing marker $k$'s cumulative exposure by $\Delta$ multiplies the hazard by $\exp(\alpha_k\Delta)$. A positive $\alpha_k$ corresponds to a higher hazard for greater cumulative exposure; a negative $\alpha_k$ corresponds to a lower hazard.

If marker $k$ is measured in units $U_k$ and time is measured in months, $A_{ik}$ has units $U_k$-months. Thus $\alpha_k$ has reciprocal units, $1/(U_k\text{-months})$, so that $\alpha_k A_{ik}$ is dimensionless. The hazard itself has units of inverse time, such as events per month. These units help distinguish the cumulative-exposure coefficient from a coefficient for a current marker value or a marker slope.

Computing Exposure From a Linear Trajectory

In a linear mixed model, a subject-specific marker trajectory often has the form $m_{ik}(s)=a_{ik}+b_{ik}s$, where $a_{ik}$ and $b_{ik}$ may depend on fixed effects, covariates, and random effects. For a full window with $t\geq W$, integration gives

\begin{aligned} A_{ik}(t;W) &=\int_{t-W}^{t}(a_{ik}+b_{ik}s)\,ds\\ &=a_{ik}W+\frac{b_{ik}}{2}\left[t^2-(t-W)^2\right]\\ &=Wm_{ik}(t)-\frac{b_{ik}W^2}{2}. \end{aligned}

The last equality follows because $m_{ik}(t)=a_{ik}+b_{ik}t$. It shows that cumulative exposure over a full window is the window length times the endpoint level, minus a correction determined by the slope. Equivalently, the integral is the window length multiplied by the average marker level over that window. For a linear trajectory, that average equals the midpoint of the two endpoint values.

This calculation also clarifies why cumulative exposure is not simply the current marker value multiplied by $W$: that shortcut is correct only for a constant trajectory over the window. If $t<W$, the lower integration limit is zero, and the same integration procedure applies over $[0,t]$, not over a nonexistent full window.

Worked Example: Integrating Two Marker Trajectories

Suppose time is measured in months and, over the interval from month 6 to month 10, a hypothetical subject has latent trajectories $m_{i1}(s)=2+0.1s$ and $m_{i2}(s)=5-0.2s$. Calculate each marker's cumulative exposure for the four-month window ending at month 10.

For both markers, $W=4$ and the interval is $[10-4,10]=[6,10]$. For marker 1,

\begin{aligned} A_{i1}(10;4) &=\int_6^{10}(2+0.1s)\,ds\\ &=2(10-6)+0.05(10^2-6^2)\\ &=8+0.05(64)\\ &=11.2\ \text{marker units}\cdot\text{month}. \end{aligned}

For marker 2,

\begin{aligned} A_{i2}(10;4) &=\int_6^{10}(5-0.2s)\,ds\\ &=5(10-6)-0.1(10^2-6^2)\\ &=20-0.1(64)\\ &=13.6\ \text{marker units}\cdot\text{month}. \end{aligned}

The two integrals are positive even though marker 2 is declining: its trajectory remains positive throughout the interval. Each result is an area under a latent trajectory, not a sum of the subject's observed readings.

Worked Hazard Calculation

Once each exposure has been calculated, its contribution to the log hazard is additive. The corresponding hazard ratio for a change in one exposure is obtained by exponentiating that coefficient times the change, while holding the other model terms fixed.

Worked Example: Combining Two Exposure Terms

Consider a hypothetical subject at month 10 with a four-month window. Suppose the latent trajectories on $[6,10]$ are $m_{i1}(s)=3$ and $m_{i2}(s)=2.825+0.1s$. Let the marker association coefficients be $\alpha_1=0.030$ per marker-unit-month and $\alpha_2=0.016$ per marker-unit-month. For this calculation, take the baseline hazard to be $0.020$ per month and set the covariate contribution to zero.

The marker 1 exposure is constant across the interval:

A_{i1}(10;4)=\int_6^{10}3\,ds=3(4)=12\ \text{marker units}\cdot\text{month}.

For marker 2,

\begin{aligned} A_{i2}(10;4) &=\int_6^{10}(2.825+0.1s)\,ds\\ &=2.825(4)+0.05(10^2-6^2)\\ &=11.3+3.2\\ &=14.5\ \text{marker units}\cdot\text{month}. \end{aligned}

The combined exposure contribution to the log hazard is

\alpha_1A_{i1}+\alpha_2A_{i2} =(0.030)(12)+(0.016)(14.5) =0.360+0.232 =0.592.

Therefore, the conditional hazard is

h_i(10\mid\mathbf b_i) =0.020\exp(0.592) \approx 0.03615 \approx 0.0362\ \text{per month}.

For these specified covariates and random effects, the hazard at month 10 is about $1.808$ times the baseline hazard, since $\exp(0.592)\approx1.808$. This is a conditional hazard comparison at that time; it is not a statement that the probability of an event during the next month is $0.0362$. A hazard is an instantaneous rate, not an interval probability.

Window Boundaries and Interpretation

The boundary rule has direct consequences for the meaning of the exposure. Under the truncated-window definition, the integration interval has length $t$ before $W$, and length $W$ thereafter. Early exposure values are therefore smaller in duration as well as potentially different in marker level. If the scientific target is a full fixed-duration exposure, one option is to define the hazard contribution only for $t\geq W$; another is to specify how pre-baseline marker history is obtained. Simply pretending that the observed interval has length $W$ changes the exposure being modeled.

Worked Example: Exposure Before a Full Window Is Available

Suppose a hypothetical marker follows $m_i(s)=2+0.5s$ from baseline at month 0, and the chosen window length is $W=4$ months. At month 3, the truncated-window definition integrates from month 0 to month 3:

\begin{aligned} A_i(3;4) &=\int_0^3(2+0.5s)\,ds\\ &=2(3)+0.25(3^2)\\ &=6+2.25\\ &=8.25\ \text{marker units}\cdot\text{month}. \end{aligned}

The observed history lasts three months, not four. Dividing by three gives an average marker level of $8.25/3=2.75$ marker units over the observed interval. Dividing by four instead would give $2.0625$, which is not the average marker level on $[0,3]$; it treats the unavailable month as if it contributed zero exposure. If a full four-month window is required, the model must instead specify how to handle the pre-baseline interval, or defer use of that exposure until month 4.

Assumptions, Checks, and Alternatives

The exposure-hazard specification is useful only when its assumptions align with the data and scientific question. In particular, the latent marker trajectories must be adequately represented over the relevant intervals, the window length and boundary rule must be justified, and the conditional proportional-hazards form must be a reasonable description of the association. The interpretation also presumes the exposure integral is based on the modeled latent trajectory rather than a selectively observed subset of measurements.

Practical diagnostics should focus on both the trajectory and hazard components. Examine whether fitted longitudinal curves track the observed marker histories, including their timing and within-subject patterns. Check that estimated trajectories do not imply implausible levels between visits or in portions of the window with little data. For the survival component, inspect residual or time-varying association diagnostics appropriate to the fitted model. Compare the chosen window with scientifically defensible alternatives as a sensitivity analysis; materially different conclusions across plausible windows indicate that the estimated association depends on the exposure definition.

Cumulative exposure answers a different question from other marker-to-hazard links. A current-value term asks whether the latent level at time $t$ relates to the hazard at $t$; a slope term asks whether the local rate of change matters. An average exposure divides the integral by the duration, giving units of marker level rather than marker-level-times-time. A weighted burden applies a time-varying weighting function within the exposure interval. These alternatives should be selected for the scientific estimand, not because one formulation is universally preferable.

With multiple markers, each cumulative term has its own coefficient, and each coefficient is interpreted conditional on the other included terms. If marker trajectories are strongly related, the data may contain limited information to distinguish their separate conditional associations. This can produce imprecise estimates even when the overall set of markers is predictive. Cumulative exposures also compress an entire history into one number: histories with different timing can have the same integral. A model using a single integral cannot distinguish those histories unless its exposure definition includes additional temporal structure.

Common Mistakes

Using observed readings as though they were the exposure integral. An irregularly spaced set of measurements is not itself an area under the latent trajectory. Compute the integral from the fitted trajectory, accounting for the modeled time course.
Ignoring the early-window boundary. Before $t=W$, the stated truncated-window exposure covers only $[0,t]$. Do not describe it as a full $W$-month exposure.
Interpreting $\alpha_k$ as a hazard ratio by itself. A change of $\Delta$ exposure units has hazard ratio $\exp(\alpha_k\Delta)$, not $\alpha_k$.
Calling the hazard an event probability. A hazard is an instantaneous rate. A probability over a time interval requires integrating the hazard, with the relevant survival model.
Assuming equal cumulative exposures imply equal histories. The integral preserves total area but not the order or timing of marker levels within the window.

Key takeaway: Define each marker's window and boundary rule first, integrate its latent trajectory to obtain $A_{ik}(t;W)$, and then include the marker-specific terms in the conditional hazard. Interpret each coefficient through a specified change in exposure, with units and the other model terms held fixed.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A latent marker is constant at 4 units over a full five-month window. What is its cumulative exposure, and what are its units?
For a full window of length $W$, explain why a linear trajectory's integral equals $W$ times the average of its endpoint levels.
If $\alpha_k=0.02$ per marker-unit-month, what is the hazard ratio associated with an increase of 10 marker-unit-months in $A_{ik}$, holding other terms fixed?
A four-month rolling window is specified, but a subject has only three months of follow-up. Under the truncated-window definition, what interval is integrated, and why is dividing its integral by four not the observed-interval average?

Solutions. (1) The exposure is $4\times5=20$ marker units-month. (2) A linear function's average over an interval is the mean of its endpoint values; multiplying that average by the interval length gives the area under the line, so the integral is $W$ times that average. (3) The hazard ratio is $\exp(0.02\times10)=\exp(0.2)\approx1.221$, or about a 22.1% higher hazard for that exposure increase. (4) Integrate over $[0,3]$, the observed interval from baseline to month 3. Dividing by four treats the missing fourth month as zero contribution; the observed-interval average is the integral divided by three.